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Negative refraction and Left-handed behavior in Photonic Crystals: FDTD and Transfer matrix method studies

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Negative refraction and Left-handed behavior in Photonic Crystals:

FDTD andTransfer matrix method studies

Peter Markos, S. Foteinopoulou and C. M. Soukoulis

- What are metamaterials?
- Historical review Left-handed Materials
- Results of the transfer matrix method
- Determination of the effective refractive index
- Negative n and FDTD results in PBGs (ENE & SF)
- New left-handed structures
- Experiments on negative refractions (Bilkent)
- Applications/Closing Remarks

E. N. Economou & S. Foteinopoulou

What is an ElectromagneticMetamaterial?

- A composite or structured material that exhibits properties not found in naturally occurring materials or compounds.
- Left-handed materials have electromagnetic properties that are distinct from any known material, and hence are examples of metamaterials.

Example: Metamaterials based on repeated cells…

(-,+)

(+,+)

(-,-)

(+,-)

Veselago

We are interested in how waves propagate through various media, so we consider solutions to the wave equation.

Sov. Phys. Usp. 10, 509 (1968)

Left-Handed Waves

- If then is a right set of vectors:
- If then is a left set of vectors:

- Energy flux (Pointing vector):
- Conventional (right-handed) medium
- Left-handed medium

Frequency dispersion of LH medium

- Energy density in the dispersive medium
- Energy density Wmust be positive and thisrequires
- LH medium is always dispersive
- According to the Kramers-Kronig relations –

it isalways dissipative

PBGs as Negative Index Materials (NIM)

- Veselago : Materials (if any) withe< 0 and m< 0
- ·em> 0 Propagation
- k, E, H Left Handed (LHM) S=c(E x H)/4p

opposite to k

- · Snell’s law with < 0 (NIM)
- ·g opposite to k
- ·Flat lenses
- ·Super lenses

S2

O΄Μ

A

B

Ο

Objections to the left-handed ideas

Parallel momentum is not conserved

Causality is violated

Fermat’s Principle ndl minimum (?)

Superlensing is not possible

- Photonic crystals have practically zero absorption
- Momentum conservation is not violated
- Fermat’s principle is OK
- Causality is not violated
- Superlensing possible but limited to a cutoff kc or 1/L

is imaginary

- Wave components with decay, i.e. are lost , then Dmax l

If n < 0, phase changes sign

if

imaginary

thus

ARE NOT LOST !!!

Metamaterials Extend Properties

J. B. Pendry

First Left-Handed Test Structure

UCSD, PRL 84, 4184 (2000)

Split rings alone

e<0

m<0

m<0

m>0

m>0

m>0

m>0

e<0

e<0

e<0

Wires alone

Transmission Measurements

Transmitted Power (dBm)

6.0

6.5

5.5

7.0

5.0

4.5

Frequency (GHz)

UCSD, PRL 84, 4184 (2000)

UCSD, APL 78, 489 (2001)

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Measurement of Refractive Index

UCSD, Science 292, 77 2001

Transfer matrix is able to find:

- Transmission (p--->p, p--->s,…)p polarization
- Reflection (p--->p, p--->s,…)s polarization
- Both amplitude and phase
- Absorption

Some technical details:

- Discretization: unit cell Nx x Ny x Nz : up to 24 x 24 x 24
- Length of the sample: up to 300 unit cells
- Periodic boundaries in the transverse direction
- Can treat 2d and 3d systems
- Can treat oblique angles
- Weak point: Technique requires uniform discretization

EM wave propagates in the z -direction

Periodic boundary conditions

are used in transverse directions

Polarization: p wave: E parallel to y

s wave: E parallel to x

For the p wave, the resonance frequency

interval exists, where with Re meff <0, Re eeff<0 and Re np <0.

For the s wave, the refraction index ns = 1.

Typical size of the unit cell: 3.3 x 3.67 x 3.67 mm

Typical permittivity of the metallic components: emetal = (-3+5.88 i) x 105

SRR

EM waves propagate in the z-direction.

Periodic boundary conditions are used in the xy-plane

LHM

Left-handed material: array of SRRs and wires

Resonance frequency as a function of metallic permittivity

complex em

Real em

Dependence of LHM peak on metallic permittivity

The length of the system is 10 unit cells

Example of Utility of Metamaterial

The transmission coefficient is an example of a quantity that can be determined simply and analytically, if the bulk material parameters are known.

UCSD and ISU, PRB, 65, 195103 (2002)

Effective permittivity e(w) and permeability m(w) of wires and SRRs

UCSD and ISU, PRB, 65, 195103 (2002)

Effective permittivity e(w) and permeability m(w) of LHM

UCSD and ISU, PRB, 65, 195103 (2002)

Effective refractive index n(w) of LHM

UCSD and ISU, PRB, 65, 195103 (2002)

Determination of effective parameters from transmission studies

From transmission and reflection data, the index of refraction n was calculated.

Frequency interval with Re n<0 and very small Im n was found.

Another 1D left-handed structure:

Both SRR and wires are located on the same side of the dielectric board.

Transmission depends on the orientation of SRR.

Bilkent & ISU APL 2002

w

t»w

t=0.5 or 1 mm

w=0.01 mm

t

0.33 mm

3 mm

l=9 cm

0.33 mm

3 mm

ax

Periodicity:

ax=5 or 6.5 mm

ay=3.63 mm

az=5 mm

Number of SRR

Nx=20

Ny=25

Nz=25

Polarization: TM

y

E

x

y

x

B

z

Phase and group refractive index

- In both the LHM and PC literature there is still a lot of confusion regarding the phase refractive index np and the group refractive index ng. How these properties relate to “negative refraction” and LH behavior has not yet been fully examined.
- There is controversy over the “negative refraction” phenomenon. There has been debate over the allowed signs (+ /-) for np and ng in the LH system.

DEFINING phase and group refractive index np and ng

- In any general case:
- The equifrequency surfaces (EFS) (i.e. contours of constantfrequency in 2D k-space) in air and in the PC are needed to find the refracted wavevector kf (see figure).
- vphase=c/|np| and vgroup= c/|ng|
- Where c is the velocity of light
- So from k// momentum conservation:|np|=c kf () /.

Remarks

- In the PC system vgroup=venergy so |ng|>1. Indeed this holds !
- np <1 in many cases, i.e. the phase velocity is larger than c in many cases.
- npcan be used in Snell’s formula to determine the angle of the propagating wavevector. In general this angle is not the propagation angle of the signal. This angle is the propagation angle of the signal only when dispersion is linear (normal), i.e. the EFS in the PC is circular (i.e. kf independent of theta).
- ng can never be used in a Snell-like formulato determine the signals propagation angle.

Index of refraction of photonic crystals

- The wavelength is comparable with the period of the photonic crystal
- An effective medium approximation is not valid

Equifrequency surfaces

Effective index

Refraction angle

kx

ky

w

Incident angle

Photonic Crystals with negative refraction.

S. Foteinopoulou, E. N. Economou and C. M. Soukoulis

Schematics for Refraction at the PC interface

EFS plot of frequency a/l = 0.58

Schematics for Refraction at the PC interface

EFS plot of frequency a/l = 0.535

Photonic Crystals

Lattice constant=4.794 mm

Dielectric constant=9.73

r/a=0.34, square lattice

Experiment by Ozbay’s group

Band structure, negative refraction and experimental set up

Frequency=13.7 GHz

Negative refraction is achievable

in this frequency range for

certain angles of incidence.

Bilkent & ISU

Subwavelength Resolution in PC based Superlens

The separation between the two point sources is l/3

Photonic Crystals with negative refraction.

Photonic

Crystal

vacuum

FDTD simulations were used to study the time evolution of an EM wave as it hits the interface vacuum/photonic crystal.

Photonic crystal consists of an hexagonal lattice of dielectric rods

with e=12.96. The radius of rods is r=0.35a. a is the lattice constant.

Photonic Crystals: negative refraction

The EM wave is trapped temporarily at the interface and after a long time,

the wave front moves eventually in the negative direction.

Negative refraction was observed for wavelength of the EM wave

l= 1.64 – 1.75 a (a is the lattice constant of PC)

- Simulated various structures of SRRs & LHMs
- Calculated transmission, reflection and absorption
- Calculated meff and eeffand refraction index(with UCSD)
- Suggested new designs for left-handed materials
- Found negative refraction in photonic crystals
- A transient time is needed for the wave to move along the - direction
- Causality and speed of light is not violated.
- Existence of negative refraction does not guarantee the existence of
- negative n and so LH behavior
- Experimental demonstration of negative refraction and superlensing
- Image of two points sources can be resolved by a distance of l/3!!!

$$$ DOE, DARPA, NSF, NATO, EU

- P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 033401 (2002)
- P. Markos and C. M. Soukoulis, Phys. Rev. E 65, 036622 (2002)
- D. R. Smith, S. Schultz, P. Markos and C. M. Soukoulis, Phys. Rev. B 65, 195104 (2002)
- M. Bayindir, K. Aydin, E. Ozbay, P. Markos and C. M. Soukoulis, Appl. Phys. Lett. (2002)
- P. Markos, I. Rousochatzakis and C. M. Soukoulis, Phys. Rev. E 66, 045601 (R) (2002)
- S. Foteinopoulou, E. N. Economou and C. M. Soukoulis, PRL, accepted (2003)
- S. Foteinopoulou and C. M. Soukoulis, submitted Phys. Rev. B (2002)
- P. Markos and C. M. Soukoulis, submitted to Opt. Lett.
- E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou and C. M. Soukoulis, submitted to Nature
- P. Markos and C. M. Soukoulis, submitted to Optics Express

The keen interest to the topic

- Left-Handed Medium (LH)
- Metamaterial
- Backward Medium (BW)
- Double Negative Medium (DNG)
- Negative Phase Velocity (NPV)
- Materials with Negative Refraction (MNR)

- Terminology

Dependence on the incident angle

Transmission peak does not depend

on the angle of incidence !

Transition peak strongly depends

on the angle of incidence.

This structure has an additional

xz - plane of symmetry

Transmission depends on the orientation of SRR

Transmission properties depend on the orientation of the SRR:

- Lower transmission
- Narrower resonance interval
- Lower resonance frequency

- Higher transmission
- Broader resonance interval
- Higher resonance frequency

Dependence of the LHM T peak on the Im eBoard

In our simulations, we have:

Periodic boundary condition,

therefore no losses due to

scattering into another direction.

Very high Im emetal therefore

very small losses in the metallic

components.

Losses in the dielectric board are crucial for the transmission

properties of the LH structures.

Superprism Phenomena in Photonic Crystals Experiment

- H.Kosaka, T.Kawashima et. al. Superprism phenomena in photonic crystals, Phys. Rev. B 58, 10096 (1998)

Scattering of the photonic crystalHexagonal 2D photonic crystal

- M.Natomi, Phys. Rev. B 62, 10696 (2000)
- Using an equifrequency surface (EFS) plots

Vanishingly small index modulation

Small index modulation

Photonic crystal as a perfect lens

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry

Phys. Rev. B, 65, 201104 (2002)

Resolution limit 0.67 l

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